ELECTRONIC
COMMUNICATIONS in PROBABILITY
EXCITED RANDOM WALK
ITAI BENJAMINI
Weizmann Institute, Rehovot 76100, Israel email: itai@wisdom.weizmann.ac.il
DAVID B. WILSON
Microsoft Research, One Microsoft Way, Redmond, WA 98052, U.S.A. email: dbwilson@microsoft.com
Submitted February 21, 2003 , accepted in final form June 10, 2003 AMS 2000 Subject classification: 60J10
Keywords: Perturbed random walk, transience
Abstract
A random walk onZdis excited if the first time it visits a vertex there is a bias in one direction,
but on subsequent visits to that vertex the walker picks a neighbor uniformly at random. We show that excited random walk onZd is transient iffd >1.
1. Excited Random Walk
A random walk on Zd is excited (with bias ε/d) if the first time it visits a vertex it steps
right with probability (1 +ε)/(2d) (ε > 0), left with probability (1−ε)/(2d), and in other directions with probability 1/(2d), while on subsequent visits to that vertex the walker picks a neighbor uniformly at random. This model was studied heavily in the framework of perturbing 1-dimensional Brownian motion, see for instance [5, 14] and reference therein. Excited random walk falls into the notorious wide category of self-interacting random walks, such as reinforced random walk, or self-avoiding walks. These models are difficult to analyze in general. The reader should consult [4, 11, 16, 15, 1], and especially the survey paper [13] for examples. Simple coupling and an additional neat observation allow us to prove that excited random walk is recurrent only in dimension 1. The proof uses and studies a special set of points (“tan points”) for the simple random walk.
2. Recurrence in Z1
It is already known that excited random walk in Z1 is recurrent, indeed, a great deal more is
known about it [6]. But for the reader’s convenience we provide a short proof.
On the first visit to a vertex there is probabilityp >1/2 of going right and 1−pof going left, while on subsequent visits the probabilities are 1/2. Suppose that the walker is atx >0 for the first time, and that all vertices between 0 and xhave been visited. The probability that the walker goes tox+ 1 before going to 0 isp+ (1−p)(1−2/(x+ 1)) = 1−2(1−p)/(x+ 1). Multiplying over thex’s, we see that the random walk returns to 0 with probability 1.
3. Transience in Z2
The simple random walk (SRW) inZ2visits aboutn/lognpoints by timen, and if the excited
random walk (ERW) gets pushed to the rightn/logntimes, it would very quickly depart its start location and never return. But it is not clear what effect that the perturbations have on the number of visited points, and it is not obvious that the excited random walk will visit n/logndistinct points by timen.
To lower bound the number of points that the excited random walk visits, we couple it with the SRW in the straightforward way, and count the number of “tan points” visited by the SRW. We define the coupling as follows: if the SRW moves up, down, or right, then so does the ERW. If the SRW moves left, then the ERW moves left if it is at a previously visited point, and if the ERW is at a new point, it moves either left or right with suitable probabilities. At all times, they-coordinates of the SRW and ERW are identical.
To explain the concept of a “tan point”, we imagine that the simple random walker leaves behind an opaque trail, and that the sun is shining from infinitely far away in the positive x-direction. If the SRW visits a point (x, y) such that no point (x′
, y) with x′
> x has been visited, then the sun shines upon (x, y), and this point becomes tanned. Formally, we define a tan point for the SRW to be a vertex (x, y) that is visited by the SRW before any point of the form (x′, y) with x′ > x. If the sun shines upon the simple random walker the first time it is at (x, y), it is straightforward to check that ERW is at a new point. We will show that with high probability there are many tan points (so the ERW visits many new points), and that this implies that the ERW is transient.
The probability that a point (x, y) will be tan follows from some enumerative work of Bousquet-M´elou and Schaeffer on random walks in the slit plane [3].
Lemma 1. Let r andθ be the polar coordinates of the point (x, y), i.e. r≥0, 0≤θ < 2π, x=rcosθ, andy=rsinθ. Then
Pr[(x, y)is tan] = (1 +o(1)) s
1 +√2 2π
sin(θ/2) √
r , (1)
where theo(1) term goes to0 asr tends to∞.
This equation does not explicitly appear in [3], but all the real work that goes into proving it is in [3]. In the interest of completeness, we explain how this equation follows from explicit results in [3]:
Proof. Letan be the number of walks of lengthn that start from (0,0), and avoid the
non-negative real axis at all subsequent times, and letpx,y,ndenote the probability that a random
such walk ends at the point (x, y). By reversibility of the random walks,
Pr "
SRW started from the point (x, y) first hits the nonnegative real axis at the point (0,0) and at timen #
= an
4n ×px,y,n.
Thus Pr[(x, y) is tan] =P∞
n=0an/4n×px,y,n. Theorem 1 of [3] gives
an
4n = (1 +o(1))
p 1 +√2 2Γ(3/4) n
−1/4.
Theorem 21 of [3] considers the endpoint (Xn, Yn) of a random walk started from (0,0) which
avoids the nonnegative real axis, and gives the limiting distribution of (Xn/√n, Yn/√n). This
would determine the asymptotics — and the authors prove a local limit theorem forYn/√n
but notXn/√nlet alone the joint distribution (Xn/√n, Yn/√n). However, since the ordinary
random walk has a local limit theorem (on vertices such thatx+y≡n mod 2), one can take the limiting distribution of X(1−ε)n, Y(1−ε)n and then run the walk another εn steps; upon
sendingε to 0 sufficiently slowly, one can obtain a local limit theorem version of Theorem 21 of [3]:
Next we check that the terms whenn≪r2 or n≫r2 contribute negligibly. We may bound an
4n×px,y,nby the probability that the walk survives the firstn/2 steps (O(1/n
1/4)) times the
probability that ordinary SRW for the remaining n/2 steps ends at the point (x, y) (O(1/n)). Thus
4n ×px,y,n by the probability that the walk makes it out to radius
r/2 without hitting the line (which isO(1/r1/2) by [11, Eqn 2.40]) times the probability that
the walk ends up at (x, y) (distances≥r/2 away) at the end of the remainingm < nsteps.
contribute negligibly, so the formula follows in this case. But the formula in the limiting case when θapproaches 0 or 2πfollows from the case whenθ is bounded away from 0 and 2π, so
the formula is valid simply when ris large enough. ¤
We will use the notation introduced by Knuth where Θ(f) denotes an expression which is upper-bounded byC×f and lower-bounded by c×f, wherec andC are positive constants. (By contrast,O(f) denotes an expression for which there is an upper bound ofC×f, but not necessarily any lower bound.) Using this notation, we may crudely approximate Equation (1) with
Pr[(x, y) is tan] = (
Θ(1/r1/2) whenx≥0
Θ(|y|/r3/2) whenx≤0. (2)
of times that the ERW is pushed to the right. But a lower bound on the expected number is not quite what we need to prove transience of the ERW; what we’d really like to know is that with very high probability, the number of tan points by timenis large.
In order to get the “with very high probability” part of the statement, it would be convenient to be working with independent events. To get this independence, we divide the plane into bands of heighth=h(n) to be determined later, and we will focus on every other band, say the even ones. When the SRW first arrives at an even band, let us count the tan points within the band that are encountered before the random walk reaches a different even band. These counts are independent for the different bands. Afternsteps it is likely that order √n/heven bands have been crossed, so it is likely that the number of tan points dominates a sum of Θ(√n/h) independent random variables.
Lemma 2. Consider a band of heighth, i.e.,Z×[y0, y0+h−1]. After the SRW first reaches
this band, with probabilityΘ(1)the SRW hitsΘ(h3/2)tan points within the band before leaving
the enclosing band Z×[y0−h, y0+ 2h−1].
For the Θ(1) part of this lemma, the following proposition is useful.
Proposition 3. IfX is a real-valued random variable andE[X]≥0, then
Pr£
This proposition is essentially exercise 1.3.8 of Durrett [8].
Proof of Lemma 2. Consider a point (x, y) within the band, and the ray [x,∞)×ywith (x, y) at its tip. Consider the largest circle contained within the enclosing band and centered at the tip, and also a small enough disk centered at the tip. From Equation (2) it follows that we can take the ratios of the two radii to be Θ(1) and have the property that for any pointpin the left half of the small disk and any pointq on the outer circle,
Pr[SRWphits ray at tip]≥2 Pr[SRWq hits ray at tip],
where SRWp denotes the simple random walk started at pointp, and by “hits ray at tip” we
mean that the first time the walker hits the ray (x+, y) is at the tip (x, y). Now
The point where the random walk first enters the band will lie within the left half of the small disk surrounding Θ(h2) such points (x, y). In fact, there are Θ(h2) such points (x, y) within
radiushof where the SRW first hits the band. Thus
E
·# tan points within band before SRW departs enclosing band and within radius hfrom where SRW arrives in band ¸
Next we need a second moment estimate:
E
"µ
# tan points within band before SRW departs enclosing band and within radiushfrom where SRW arrives in band
¶2#
≤Eh( # tan points within radiushfrom where SRW arrives in band)2i
= X
x1,y1,x2,y2within radiush
Pr[(x1, y1) and (x2, y2) tan]
≤2 X
x1,y1,x2,y2within radiush
Pr[(x1, y1) tan and (x2, y2) tan after (x1, y1)]
where by “after” we include the possibility that (x2, y2) = (x1, y1)
= 2X Pr[(x1, y1) tan]×Pr[(x2, y2) after (x1, y1)|(x1, y1) tan]×
Pr[(x2, y2) tan|(x2, y2) after (x1, y1) and (x1, y1) tan]
≤2XPr[(x1, y1) tan]×Pr
"
after (x1, y1), (x2, y2) visited before (x2+, y2)|
(x2, y2) after (x1, y1) and (x1, y1) tan
#
= 2XPr[(x1, y1) tan] Pr[(x2−x1, y2−y1) tan in SRW0]
= Θ(h3)
Combining these estimates with Proposition 3, we see that with at least Θ(1) probability there are at least Θ(h3/2) tan points in the band before the SRW departs the enclosing band. ¤
Theorem 4. With probability1, for all but finitelyn∈N, the excited random walk has drifted
right by a distance of at least Θ(n3/4/log5/4n)
at timen. In particular, it is transient. Proof. Say that the SRW deals with a band if it reaches that band and then reaches a different band of the same parity. Suppose that the random walk starts from an odd band in the middle of a group of 4k+ 1 bands of heighth. Then the probability that it fails to leave the group of bands after n= (kh)2t steps is exponentially small int. We will optimize kand t later, but
we will take t to be large so that with high probability the walk leaves the group of 4k+ 1 bands, and in particular deals with at least keven bands.
Rather than run the random walk for exactlynsteps, let us run it until it deals with k even bands. Then the number of early tan points in the different even bands are independent of one another, and each one has a Θ(1) chance of being at least Θ(h3/2). Except with probability
exp(−Θ(k)), the number of tan points in the keven bands will be Θ(kh3/2).
Since there is only a exp(−Θ(t)) chance that the walk has not dealt with k even bands by time n, we find that, except with probability exp(−Θ(t)) + exp(−Θ(k)), there are Θ(kh3/2)
tan points by timen. To optimize our parameters we taket=k, and then we haven=k3h2.
Next we consider the location of the perturbed random walk at timen. Typically the random walk diffuses by Θ(√n) = Θ(k3/2h), and drifts right by at least Θ(kh3/2). The probability
that it diffuses by more than k2his exp(−Θ(k)), and the probability that it drifts less than
Θ(kh3/2) is <exp(−Θ(k)). We takek = Θ(logn), so that the probability of a bad event is
< 1/n2, which gives us h= Θ(n1/2/log3/2n). Except with probability <1/n2, the excited
random walk has drifted right by at least Θ(n3/4/log5/4
n). In particular this event fails only
4. Transience in Zd, d >2
As with ERW inZ2, inZd we can couple the ERW with SRW, and then (in continuous time)
couple the SRW inZd with the SRW inZ2. For each tan point of the SRW inZ2 there is a
tan point of the SRW inZd, so as before the ERW is transient.
5. Speed
We have seen that the excited random walk on Zd is transient for d ≥2, but does it have
positive speed?
Theorem 5. Let Xn denote thex-coordinate at timenof the excited random walk onZd with
bias ε/d. Ifd≥4, then almost surelylim infn→∞Xn/n≥0.659ε/d; in particular the speed is
positive.
Proof. Project down the x-coordinate of the ERW and d−4 additional coordinates, and consider the resulting SRW onZ3. LetRn be the range of the SRW onZ3by timen, i.e. the
number of points visited by time n. Since the SRW is transient,E[Rn]/n→cwhere c is the
escape probability of the SRW. Glasser and Zucker [10] (see also [7]) determined this escape probabilityc to be
c= 32π
3
√
6Γ(1/24)Γ(5/24)Γ(7/24)Γ(11/24) = 0.65946. . . .
For our purposes it is not enough to know E[Rn], what we need is the strong law of large
numbers for Rn that was proved by Dvoretzky and Erd˝os [9]: a.s.Rn/n→c(see also [2] for
even stronger results onRn). Thus for anyδ >0, a.s. there are only finitely manynfor which
the ERW has not had (c−δ)npushes to the right by timen. The theorem then follows from
the ordinary strong law of large numbers. ¤
It seems intuitive that excited random walk inZ3 also has positive speed, but we do not see
a proof. Excited random walk inZ2 is more delicate, and it is not clear even at an intuitive
level whether or not the speed is positive, though we believe that by time n it has traveled distance at least Θ(n/logn).
Acknowledgements
We are grateful to Oded Schramm for useful discussions, and we thank Gabor Pete and the referee for their comments on an earlier version of this article. The research leading to this article was conducted while the first author was visiting Microsoft.
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